If you are using MobileMe to push content to your iPhone it will push from the web interface, so you won't get any CalDav or subscription content.

However, apparently (I haven't tried this) you can have multiple "push" email accounts (say MobileMe and Zimbra) and allow both to push data to the calendar on the iPhone. You can't combine push and iTunes sync, but you could push from MobileMe and a Zimbra server (but again, that's only what I've read).

Yes, the open source version supports caldav, so you can use that plus iTunes sync to get Zimbra events onto the iPhone's calendar.

You can also access the Zimbra web interface from the iPhone and it will provide a web site tuned to the iPhone. However, it doesn't allow editing or creation of events yet (though that seems to be planned).